Abstract

We study the existence of ground state solutions of the periodic discrete coupled nonlinear Schrödinger lattice by using the Nehari manifold approach combined with periodic approximations. We show that both of the components of the ground state solutions are not zero.

1. Introduction

In this paper, we consider the coupled discrete Schrödinger system
where is a positive constant, is a real valued -periodic sequence, , and . is the discrete Laplacian operator defined as .

System (1) could be viewed as the discretization of the two-component system of time-dependent nonlinear Gross-Pitaevskii system (see [1] for more detail) as

It is well known that coupled nonlinear Schrödinger equations arise quite naturally in nonlinear optics [2] and Bose-Einstein condensates. Bose-Einstein condensation for a mixture of different interaction atomic species with the same mass was realized in 1997 (see [3]), which stimulated various analytical and numerical results on the ground state solutions of system (2). The discrete nonlinear Schrödinger equations (DNLS) have a crucial role in the modeling of a great variety of phenomena, ranging from solid-state and condensed-matter physics to biology. During the last years, there has been a growing interest in approaches to the existence problem for ground states. We refer to the continuation methods in [4, 5], which have been proved to be powerful for both theoretical considerations and numerical computations (see [6]), to [7], which exploits spatial dynamics and centre manifold reduction, to the variational methods in [8–15], which rely on critical point techniques (linking theorems, the Nehari manifold), and to the Krasnoselskii fixed point theorem together with a suitable compactness criterion [16].

The aim of this paper is to study the discrete solitons of (1), that is, solutions of the form
where the amplitudes and are supposed to be real. Inserting the ansatz of the discrete solitons (3) into (1), we obtain the following equivalent algebraic equations:
and (4) becomes

In fact, we consider the following more general equations:
where and are the second-order difference operators defined by
where , , , and are real valued -periodic sequences, and , , and are positive numbers. Obviously, (5) is a special case of (7) with , , , and .

Since, for , the operator is a bounded and self-adjoint operator in , its spectrum has a band structure; that is, is a union of a finite number of closed intervals [17]. The complement consists of a finite number of open intervals called spectral gaps and two of them are semi-infinite which are denoted by and , respectively.

In this paper, we consider two types of solutions to (7) as follows: (i) -periodic, that is, , , and (ii) discrete solitons. Actually, in case (ii), we look for solutions in the space ; then (6) holds naturally. System (7) has a trivial solution , . We are looking for nontrivial solutions.

The main idea in this paper is as follows. First, we consider (7) in a finite -periodic sequence space, and is not a spectrum of the corresponding operator , . By using the Nehari manifold approach, we obtain the existence of -periodic solutions. Then we show that these solutions have upper and lower bounds. Finally, by an approximation technique, we prove that the limit of these solutions exists and is the solution of (7) in . Compared with the existence of ground state solutions of the DNLS, the difficulty is that we need to show that both of the components of the ground state solutions are not zero.

The remaining of this paper is organized as follows. First, in Section 2, we establish the variational framework associated with (7) and introduce the Nehari manifolds. Then, in Section 3, we present a sufficient condition on the existence of -periodic solutions and nontrivial solutions in of (7).

2. Preliminaries and the Nehari Manifold

In this section, we first establish the variational framework associated with (7).

Let be the set of the following form:

For any fixed positive integer , we define the subspace of as
Obviously, is isomorphic to and hence can be equipped with the inner product and norm as
respectively. Sometimes, we will consider norm on as
We also define a norm on by
The symbol stands for the space with the norm
and the inner product
We also consider norm on as
We mention that
where .

Consider the functionals on and on defined by
respectively. Then and its derivative is given by
and and its derivative is given by

Let be the distance from to the spectrum ; that is,
Let . Then, we have
Furthermore, we let
Then, obviously, and .

Denote
Then

Next, we study the main properties of the Nehari manifolds with the functionals and .

Let
Then and are functionals and their derivatives are given by
respectively. The Nehari manifolds are defined as follows:
Note that contains all critical points of in .

To prove the main results, we need some lemmas on the Nehari manifolds.

Lemma 1. Assume that and hold. Then the sets and are nonempty closed submanifolds in and , respectively. The derivatives and are nonzero on the corresponding Nehari manifolds. Moreover, there exists such that , and , .

Proof. The proofs for both cases are similar. We only provide the proof for the case of as an illustration. First, we show that .Let . Then by (21)
Noticing that and , by (33), we see that for small enough and for large enough. As a consequence, there exists such that ; that is, .Let . By (21), (30), and the definition of , we have
Hence, , and the implicit function theorem implies that is a submonifold in . Now let us prove the last statement of the lemma. Let . By the assumption of Lemma 1 and the definition of , we have
where . Let . Then, by (24) and (35), it is easy to see that
which implies that
Closedness of is obvious. The proof is completed.

Lemma 2. Assume that and hold. Then there exists such that for all .

Proof. For any , we have
By Lemma 1, we know that . Hence, let . Then (38) implies that . The proof is completed.

Lemma 3. For , the function , , has a unique critical point at , which is, actually, a global maximum. The same statement holds for and .

Proof. Let , , . Computing the derivative of , we have
This shows that is a unique maximum point. The proof is completed.

Lemma 4. Let be a minimizer of the functional constrained on the Nehari manifold ; that is,
and then is a nontrivial -periodic solution to (7), which is called a nontrivial periodic ground state solution to (7).

Proof. According to Lagrange multiplier method, is the critical point of the functional . Thus , and for arbitrary ,
After taking , we obtain
But
Thus, and
for any . Take and in (44) for , where
We see that . Thus, is a nontrivial -periodic ground state solution to (7). The proof is completed.

Through a similar argument to the proof of Lemma 4, we get the following lemma.

Lemma 5. Let be a minimizer of the functional constrained on the Nehari manifold ; that is,
and then is a nontrivial discrete soliton to (7), which is called a nontrivial ground state solution to (7).

3. Main Results

In this section, we will establish some sufficient conditions on the existence of -periodic solutions and nontrivial solutions in of (7).

We start with the following.

Lemma 6. Assume that and hold. Then the minimum value in (40) is attained.

Proof. Let be a minimizing sequence for ; that is, as . The fact that
shows that is bounded. Since the space is finite dimensional, so the norm is equivalent to the Euclidean norm on , and the sequence is bounded. Passing to a subsequence, we can assume that converges to . Since the set is closed and the functional is continuous, we obtain that and . The proof is completed.

To obtain a nontrivial solutions in of (7), we need the following lemma.

Lemma 7. Assume that and hold. Let be a -periodic ground state solution, that is, a solution of (40). Then the sequences and are bounded above and below away from zero.

Proof. By Lemma 2, is obviously bounded below away from zero. Let
so , as in the beginning of the proof of Lemma 1; there exists such that ; that is, . Therefore, . And is bounded above.Next, we prove that are bounded above and below away from zero. By the previous proof, we see that is bounded above and below away from zero; that is, there exist and such that . Then
This implies that
The proof is completed.

Lemma 8. Assume that and hold. Let be a -periodic ground state solution, that is, a solution of (40). Then there exist positive constants and such that

Proof. By Lemmas 4 and 6, we know that is a nontrivial critical point of . Therefore, we have
Let with , such that and . By the fact that
we get
If one of the components of , say , is equal to 0, then . Thus, by (54), we obtain
By (56), we get
Similarly, if , then we have
If and , then, by (54) and (55), we obtain
By (59), we get
Let
Then, by (57), (58), and (60), we get (51). The proof is completed.

Now we are ready to state our main results.

Theorem 9. Assume that and hold. Then for every positive integer , (7) possesses a nontrivial -periodic ground state solution in . Moreover, there are other three nontrivial -periodic ground state solutions , , and to (7).

Theorem 10. Assume that , , , and
holds. Then system (7) has a nontrivial ground state solution in with and . Moreover, there are other three nontrivial ground state solutions , and , to (7).

Proof. Consider the sequence of -periodic solutions found in Lemma 6. By Lemma 8, without loss of generality, we can assume that the subsequence of satisfies . Thus, there exists such that
By the periodicity of the coefficients in (7), we see that is also a solution to (7). Making some shifts if necessary, without loss of generality, we can assume that in (63). Moreover, passing to a subsequence of , we can also assume that and . It follows from (37), (50), (51), and (63) that we can choose a subsequence, still denoted by and , such that and for all . Notice that . Then with . Furthermore, (7) possesses pointwise limits and hence is a nontrivial solution to (7). By the way, , , and are nontrivial solutions to (7). Since , if is a ground state solution, then , , and are the ground state solutions.Now we prove that the solution is a ground state solution. By Lemma 5, we have to show that . Actually, we have proven that, for every sequence , passing to a subsequence still denoted by and making appropriate shifts, we can suppose that and pointwise, where is a nontrivial solution. For any positive integer , we have
Let ; we obtain that
and, hence,
Now we prove that
By Lemma 7, we know that the sequence is bounded above and below away from zero. We extract a subsequence, still denoted by , and hence we prove that
Given , let be such that
Choose sufficiently close to 1 such that
We also have that . By density argument, we can find a finitely supported sequence sufficiently close to in such that and
Then there exists such that and
Let be such that and if . If is large enough, then and
This implies (68). Hence, by (66) and (68), we have .Finally, we will show that and . From the above arguments, system (7) has a nontrivial ground state solution in . If one of the components of , say , is equal to 0, then (the proof for the other case is similar). For small enough, we consider . By a similar argument to the proof of Lemma 1, we know that there exists such that ; that is . By , we have , where
Moreover, . Notice that , , and
For the sake of simplicity, we let
If , then, by (28) and (75),
This, combined with and (77) yields . If , then, by (28) and (75),
This, together with and (78) gives . Thus, if , then we have . For small enough, we have
Hence, by (79), we have
This is a contradiction. So, . The proof is completed.

Acknowledgments

The authors would like to thank the anonymous referee for his/her valuable suggestions. This work is supported by Program for Changjiang Scholars and Innovative Research Team in University (no. IRT1226), the National Natural Science Foundation of China (no. 11171078), the Specialized Fund for the Doctoral Program of Higher Education of China (no. 20114410110002), and SRF of Guangzhou Education Bureau (no. 10A012).